SCHEIBEHENNE, Benjamin, PACHUR, Thorsten. Using Bayesian hierarchical parameter estimation to assess the generalizability of cognitive models of choice. Psychonomic Bulletin

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Article

Reference

Using Bayesian hierarchical parameter estimation to assess the generalizability of cognitive models of choice

SCHEIBEHENNE, Benjamin, PACHUR, Thorsten

Abstract

To be useful, cognitive models with fitted parameters should show generalizability across time and allow accurate predictions of future observations. It has been proposed that hierarchical procedures yield better estimates of model parameters than do nonhierarchical, independent approaches, because the formers' estimates for individuals within a group can mutually inform each other. Here, we examine Bayesian hierarchical approaches to evaluating model generalizability in the context of two prominent models of risky choice—cumulative prospect theory (Tversky & Kahneman, 1992) and the transfer-of-attention-exchange model (Birnbaum

& Chavez, 1997). Using empirical data of risky choices collected for each individual at two time points, we compared the use of hierarchical versus independent, nonhierarchical Bayesian estimation techniques to assess two aspects of model generalizability: parameter stability (across time) and predictive accuracy. The relative performance of hierarchical versus independent estimation varied across the different measures of generalizability. The hierarchical approach improved parameter stability [...]

SCHEIBEHENNE, Benjamin, PACHUR, Thorsten. Using Bayesian hierarchical parameter estimation to assess the generalizability of cognitive models of choice. Psychonomic Bulletin

& Review (PB&R) , 2015, vol. 22, no. 2, p. 391-407

DOI : 10.3758/s13423-014-0684-4

Available at:

http://archive-ouverte.unige.ch/unige:75430

Disclaimer: layout of this document may differ from the published version.

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Online Supplementary Material

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BUGS code for TAX (independent estimate of a single individual)

model {

#error parameter theta.pre~dbeta(1,1)

theta <- theta.pre*5 # range of theta from 0 to 5

#set TAX parameter n <- 2

d.pre ~ dbeta(1,1) #delta (configural weight)

d <- 4*d.pre-2 #range of d scaled from -2 to 2

g.pre ~ dbeta(1,1) #gamma (probability weighting) g <- g.pre * 5 #range scaled from 0 to 5

b.pre ~ dbeta(1,1) #beta (utility)

b <- b.pre * 5 #range scaled from 0 to 5

# calculate subjective probability pSubMaxA <- pow(probs[,1],g)

pSubMinA <- pow(probs[,3],g) pSubMaxB <- pow(probs[,2],g) pSubMinB <- pow(probs[,4],g)

#calculate weights

weightAmax <- pSubMaxA - d/(n+1) * pSubMinA #weight for high outcome is reduced weightAmin <- pSubMinA + d/(n+1) * pSubMaxA #weight for low outcome is increased

weightBmax <- pSubMaxB - d/(n+1) * pSubMinB weightBmin <- pSubMinB + d/(n+1) * pSubMaxB

#calculate utilities

utilMaxA <- signPayoffs[,1]*pow(abs(payoffs[,1]),b) utilMinA <- signPayoffs[,3]*pow(abs(payoffs[,3]),b) utilMaxB <- signPayoffs[,2]*pow(abs(payoffs[,2]),b) utilMinB <- signPayoffs[,4]*pow(abs(payoffs[,4]),b)

#obtain tax utilities for both options

taxA <- (weightAmax*utilMaxA+weightAmin*utilMinA)/(pSubMaxA+pSubMinA) taxB <- (weightBmax*utilMaxB+weightBmin*utilMinB)/(pSubMaxB+pSubMinB)

#differnce between tax utilities taxDiff <- taxA-taxB

for (choice in 1:nChoices) {

#Luce choice rule

pA[choice] <- 1/(1+exp(-1*theta*taxDiff[choice]))

#Likelihood function

choiceA[choice]~dbern(pA[choice]) }

}

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Data Provided from outside BUGS:

probs 138 x 4 matrix containing the probabilities for each pair of gambles A and B.

payoffs 138 x 4 matrix containing the payoffs for each pair of gambles A and B.

signPayoff 138 x 4 matrix indicating positive (1) and negative (-1) payoffs nChoices 138

choiceA vector containing the observed choices for all 138 pairs of gambles (coded 0 and 1)

Columns for probs and payoffs are sorted by the maximum payoff: maximum A, maximum B, minimum A, and maximum B

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BUGS code for TAX (hierarchical estimate for a group as a whole)

model {

#set TAX parameter n <- 2

#set group-level parameters mu.phi.d ~ dnorm(0,1)

tau.phi.d <- pow(sigma.phi.d, -2) sigma.phi.d ~ dunif(0,10)

mu.phi.g ~ dnorm(0,1)

tau.phi.g <- pow(sigma.phi.g, -2) sigma.phi.g ~ dunif(0,10)

mu.phi.b ~ dnorm(0,1)

tau.phi.b <- pow(sigma.phi.b, -2) sigma.phi.b ~ dunif(0,10)

mu.phi.theta ~ dnorm(0,1)

tau.phi.theta <- pow(sigma.phi.theta, -2) sigma.phi.theta ~ dunif(0,10)

for (i in 1:nVpn) {

#error parameter

theta.probit[i]~dnorm(mu.phi.theta,tau.phi.theta) theta.pre[i]<-phi(theta.probit[i])

theta[i]<-theta.pre[i]*5

d.probit[i] ~ dnorm(mu.phi.d,tau.phi.d) #delta (configural weight) d.pre[i] <- phi(d.probit[i])

d[i] <- 4*d.pre[i]-2 #- scale d from -2 to 2

g.probit[i] ~ dnorm(mu.phi.g, tau.phi.g)

g.pre[i] <- phi(g.probit[i]) #gamma (probability weighting) g[i] <- g.pre[i] * 5 #scale from 0 to 5

b.probit[i] ~ dnorm(mu.phi.b, tau.phi.b)

b.pre[i] <- phi(b.probit[i]) #beta (utility) b[i] <- b.pre[i] * 5 #scale from 0 to 5

for (choice in 1:nChoices[i]) {

# calculate subjective probability

pSubMaxA[i,choice] <- pow(probs[i,choice,1],g[i]) pSubMinA[i,choice] <- pow(probs[i,choice,3],g[i]) pSubMaxB[i,choice] <- pow(probs[i,choice,2],g[i]) pSubMinB[i,choice] <- pow(probs[i,choice,4],g[i])

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#calculate weights

weightAmax[i,choice] <- pSubMaxA[i,choice]

- d[i]/(n+1) * pSubMinA[i,choice] #weight for high outcome reduced

weightAmin[i,choice] <- pSubMinA[i,choice]

+ d[i]/(n+1) * pSubMaxA[i,choice] #weight for low outcome increased

weightBmax[i,choice] <- pSubMaxB[i,choice]

- d[i]/(n+1) * pSubMinB[i,choice]

weightBmin[i,choice] <- pSubMinB[i,choice]

+ d[i]/(n+1) * pSubMaxB[i,choice]

#calculate utilities

utilMaxA[i,choice] <-

signPayoffs[i,choice,1]*pow(abs(payoffs[i,choice,1]),b[i])

utilMinA[i,choice] <-

utilMaxB[i,choice] <-

utilMinB[i,choice] <-

#obtain tax utilities for both options taxA[i,choice] <-

(weightAmax[i,choice]*utilMaxA[i,choice]+weightAmin[i,choice]

*utilMinA[i,choice])/(pSubMaxA[i,choice]+pSubMinA[i,choice])

taxB[i,choice] <-

(weightBmax[i,choice]*utilMaxB[i,choice]+weightBmin[i,choice]

*utilMinB[i,choice])/(pSubMaxB[i,choice]+pSubMinB[i,choice])

#differnce between tax utilities

taxDiff[i,choice] <- taxA[i,choice]-taxB[i,choice]

#Luce choice rule

pA[i,choice] <- 1/(1+exp(-1*theta[i]*taxDiff[i,choice]))

#Likelihood function

choiceA[i,choice]~dbern(pA[i,choice]) }

} }

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Data Provided from outside BUGS:

probs 64 x 138 x 4 array containing the probabilities for each pair of gambles A and B for each participant

payoffs 64 x 138 x 4 array containing the payoffs for each pair of gambles A and B for each participant.

signPayoff 64 x 138 x 4 array indicating positive (1) and negative (-1) payoffs nChoices Vector containing the number of choices (i.e. 138) for each participant.

choiceA Vector containing the observed choices for all 138 pairs of gambles (coded 1 for choosing option A and 0 for choosing option B)

nVpn 64

Columns for probs and payoffs are sorted by the maximum payoff: maximum A, maximum B, minimum A, and maximum B

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BUGS code for CPT (independent estimate for a single individual)

model {

for (j in 1:64)# Subject-loop {

alpha[j] ~ dbeta(1,1) #range 0 to 1

luce.pre[j] ~dbeta(1,1) luce[j]<-luce.pre[j]*5 lambda.pre[j] ~dbeta(1,1)

lambda[j]<-lambda.pre[j]*5

#probability weighting

gamma.gain[j] ~dbeta(1,1) #range 0 to 1 gamma.loss[j] ~dbeta(1,1) #range 0 to 1

delta.gain.pre[j] ~dbeta(1,1) delta.loss.pre[j] ~dbeta(1,1)

delta.gain[j]<-delta.gain.pre[j]*5 delta.loss[j]<-delta.loss.pre[j]*5

for (i in 1:70)# Item-Loop - positive gambles

{

#---

#positive gambles,gamble A

v.x.a[i,j] <- pow(prospects.a[j,i,1],alpha[j])

v.y.a[i,j] <- pow(prospects.a[j,i,3],alpha[j])

#2parameter weighting function

w.x.a[i,j] <- delta.gain[j]*pow(prospects.a[j,i,2],gamma.gain[j])/

(delta.gain[j]*pow(prospects.a[j,i,2],gamma.gain[j])+pow((1- prospects.a[j,i,2]),gamma.gain[j]))

w.y.a[i,j] <- 1-w.x.a[i,j] #delta[j]*pow(prospects.a[j,i,4],gamma[j])/

(delta[j]*pow(prospects.a[j,i,4],gamma[j])+pow((1-prospects.a[j,i,4]),gamma[j])) Vf.a[i,j] <- w.x.a[i,j] * v.x.a[i,j] + w.y.a[i,j] * v.y.a[i,j]

#--- #positive gambles,gamble B

v.x.b[i,j] <- pow(prospects.b[j,i,1],alpha[j]) v.y.b[i,j] <- pow(prospects.b[j,i,3],alpha[j])

w.x.b[i,j] <- delta.gain[j]*pow(prospects.b[j,i,2],gamma.gain[j])/

(delta.gain[j]*pow(prospects.b[j,i,2],gamma.gain[j])+pow((1- prospects.b[j,i,2]),gamma.gain[j]))

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w.y.b[i,j] <- 1-w.x.b[i,j] #delta[j]*pow(prospects.b[j,i,4],gamma[j])/

(delta[j]*pow(prospects.b[j,i,4],gamma[j])+pow((1-prospects.b[j,i,4]),gamma[j]))

Vf.b[i,j] <- w.x.b[i,j] * v.x.b[i,j] + w.y.b[i,j] * v.y.b[i,j]

#---

# positive gambles,choice-rule

binval[i,j] <- (1)/(1+exp((-1*luce[j])*(Vf.b[i,j]-Vf.a[i,j]))) rawdata[i,j] ~ dbern(binval[i,j])

}

for (i in 71:100)# Item-Loop, negative gambles,gamble A {

v.x.a[i,j] <- lambda[j]*(-1) * pow((-1 * prospects.a[j,i,1]),alpha[j]) v.y.a[i,j] <- lambda[j]*(-1) * pow((-1 * prospects.a[j,i,3]),alpha[j])

w.x.a[i,j] <- 1-(delta.loss[j]*pow(prospects.a[j,i,4],gamma.loss[j])/

(delta.loss[j]*pow(prospects.a[j,i,4],gamma.loss[j])+pow(prospects.a[j,i,2],gamma.loss[

j])))

w.y.a[i,j] <- 1-w.x.a[i,j

Vf.a[i,j] <- w.x.a[i,j] * v.x.a[i,j] + w.y.a[i,j] * v.y.a[i,j]

#---

# Item-Loop, negative gambles,gamble B

v.x.b[i,j] <- lambda[j]*(-1) * pow((-1 * prospects.b[j,i,1]),alpha[j]) v.y.b[i,j] <- lambda[j]*(-1) * pow((-1 * prospects.b[j,i,3]),alpha[j])

w.x.b[i,j] <- 1-(delta.loss[j]*pow(prospects.b[j,i,4],gamma.loss[j])/

(delta.loss[j]*pow(prospects.b[j,i,4],gamma.loss[j])+pow(prospects.b[j,i,2],gamma.loss[

j])))

w.y.b[i,j] <- 1-w.x.b[i,j]

#---

# Item-Loop, negative gambles,choice-rule

}

for (i in 101:138)# Item-Loop, mixed gambles,gamble A {

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v.y.a[i,j] <- (-1 * lambda[j]) * pow((-1 * prospects.a[j,i,3]),alpha[j])

(delta.gain[j]*pow(prospects.a[j,i,2],gamma.gain[j])+pow((prospects.a[j,i,4]),gamma.gai n[j]))

w.y.a[i,j] <- delta.loss[j]*pow(prospects.a[j,i,4],gamma.loss[j])/

(delta.loss[j]*pow(prospects.a[j,i,4],gamma.loss[j])+pow((prospects.a[j,i,2]),gamma.los s[j]))

#--- # Item-Loop, mixed gambles,gamble B

v.x.b[i,j] <- pow(prospects.b[j,i,1],alpha[j])

v.y.b[i,j] <- (-1 * lambda[j]) * pow((-1 * prospects.b[j,i,3]),alpha[j])

(delta.gain[j]*pow(prospects.b[j,i,2],gamma.gain[j])+pow((prospects.b[j,i,4]),gamma.gai n[j]))

w.y.b[i,j] <- delta.loss[j]*pow(prospects.b[j,i,4],gamma.loss[j])/

(delta.loss[j]*pow(prospects.b[j,i,4],gamma.loss[j])+pow((prospects.b[j,i,2]),gamma.los s[j]))

#---

# Item-Loop, mixed gambles,choice-rule

} } }

Data Provided from Outside BUGS:

prospects.a 64 x 138 x 4 array containing the payoffs and probabilities for gamble A prospects.b 64 x 138 x 4 array containing the payoffs and probabilities for gamble B rawdata 138 x 64 matrix containing the observed choices for all 138 pairs of gambles

(0 for choosing option A and 1 for choosing option B)

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BUGS code for CPT (hierarchical estimate for a group of individuals)

model {

for (j in 1:64) {

alpha.phi[j] ~ dnorm(mu.phi.alpha,tau.phi.alpha) T(-5, 5) alpha[j] <- phi(alpha.phi[j]) #range 0 to 1

gamma.gain.phi[j] ~ dnorm(mu.phi.gamma.gain,tau.phi.gamma.gain) T(-5, 5) gamma.loss.phi[j] ~ dnorm(mu.phi.gamma.loss,tau.phi.gamma.loss) T(-5, 5)

gamma.gain[j] <- phi(gamma.gain.phi[j]) #range 0 to 1 gamma.loss[j] <- phi(gamma.loss.phi[j]) #range 0 to 1

delta.gain.phi[j] ~ dnorm(mu.phi.delta.gain,tau.phi.delta.gain) T(-5,5) delta.gain.pre[j] <- phi(delta.gain.phi[j])

delta.gain[j] <- delta.gain.pre[j]*5

delta.loss.phi[j] ~ dnorm(mu.phi.delta.loss,tau.phi.delta.loss) T(-5,5) delta.loss.pre[j] <- phi(delta.loss.phi[j])

delta.loss[j] <- delta.loss.pre[j]*5

luce.phi[j] ~ dnorm(mu.phi.luce, tau.phi.luce) T(-5,5) luce.pre[j] <- phi(luce.phi[j])

luce[j] <- luce.pre[j]*5

lambda.phi[j] ~ dnorm(mu.phi.lambda, tau.phi.lambda) T(-5,5) lambda.pre[j] <- phi(lambda.phi[j])

lambda[j] <- lambda.pre[j]*5

}

mu.phi.alpha ~ dnorm(0,1)

tau.phi.alpha <- pow(sigma.phi.alpha,-2) sigma.phi.alpha ~ dunif(0,10)

#probability weighing mu.phi.gamma.gain ~ dnorm(0,1)

tau.phi.gamma.gain <- pow(sigma.phi.gamma.gain,-2) sigma.phi.gamma.gain ~ dunif(0,10)

mu.phi.gamma.loss ~ dnorm(0,1)

tau.phi.gamma.loss <- pow(sigma.phi.gamma.loss,-2) sigma.phi.gamma.loss ~ dunif(0,10)

mu.phi.delta.gain ~ dnorm(0,1)

tau.phi.delta.gain <- pow(sigma.phi.delta.gain,-2) sigma.phi.delta.gain ~ dunif(0,10)

mu.phi.delta.loss ~ dnorm(0,1)

tau.phi.delta.loss <- pow(sigma.phi.delta.loss,-2)

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sigma.phi.delta.loss ~ dunif(0,10)

mu.phi.lambda ~ dnorm(0,1)

tau.phi.lambda <- pow(sigma.phi.lambda,-2) sigma.phi.lambda ~ dunif(0,10)

mu.phi.luce ~ dnorm(0,1)

tau.phi.luce <- pow(sigma.phi.luce,-2) sigma.phi.luce ~ dunif(0,10)

for (j in 1:64)# Subject-loop {

for (i in 1:70)# Item-Loop - positive gambles

{

#---

#positive gambles,gamble A

v.y.a[i,j] <- pow(prospects.a[j,i,3],alpha[j])

(delta.gain[j]*pow(prospects.a[j,i,2],gamma.gain[j])+pow((1- prospects.a[j,i,2]),gamma.gain[j]))

w.y.a[i,j] <- 1-w.x.a[i,j]

#--- #positive gambles,gamble B

v.x.b[i,j] <- pow(prospects.b[j,i,1],alpha[j]) v.y.b[i,j] <- pow(prospects.b[j,i,3],alpha[j])

(delta.gain[j]*pow(prospects.b[j,i,2],gamma.gain[j])+pow((1- prospects.b[j,i,2]),gamma.gain[j]))

w.y.b[i,j] <- 1-w.x.b[i,j] #delta[j]*pow(prospects.b[j,i,4],gamma[j])/

#---

# positive gambles,choice-rule

}

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for (i in 71:100)# Item-Loop, negative gambles,gamble A

{

v.x.a[i,j] <- lambda[j]*(-1) * pow((-1 * prospects.a[j,i,1]),alpha[j]) v.y.a[i,j] <- lambda[j]*(-1) * pow((-1 * prospects.a[j,i,3]),alpha[j])

w.x.a[i,j] <- 1-(delta.loss[j]*pow(prospects.a[j,i,4],gamma.loss[j])/

(delta.loss[j]*pow(prospects.a[j,i,4],gamma.loss[j])+pow(prospects.a[j,i,2],gamma.loss[

j])))

w.y.a[i,j] <- 1-w.x.a[i,j]#delta[j]*pow(prospects.a[j,i,4],gamma[j])/

(delta[j]*pow(prospects.a[j,i,4],gamma[j])+pow((1-prospects.a[j,i,4]),gamma[j]))

#---

# Item-Loop, negative gambles,gamble B

v.x.b[i,j] <- lambda[j]*(-1) * pow((-1 * prospects.b[j,i,1]),alpha[j]) v.y.b[i,j] <- lambda[j]*(-1) * pow((-1 * prospects.b[j,i,3]),alpha[j])

w.x.b[i,j] <- 1-(delta.loss[j]*pow(prospects.b[j,i,4],gamma.loss[j])/

(delta.loss[j]*pow(prospects.b[j,i,4],gamma.loss[j])+pow(prospects.b[j,i,2],gamma.loss[

j])))

w.y.b[i,j] <- 1-w.x.b[i,j]#delta[j]*pow(prospects.b[j,i,4],gamma[j])/

#---

# Item-Loop, negative gambles,choice-rule

}

for (i in 101:138)# Item-Loop, mixed gambles,gamble A

{

v.y.a[i,j] <- (-1 * lambda[j]) * pow((-1 * prospects.a[j,i,3]),alpha[j])

(delta.gain[j]*pow(prospects.a[j,i,2],gamma.gain[j])+pow((prospects.a[j,i,4]),gamma.gai n[j]))

w.y.a[i,j] <- delta.loss[j]*pow(prospects.a[j,i,4],gamma.loss[j])/

(delta.loss[j]*pow(prospects.a[j,i,4],gamma.loss[j])+pow((prospects.a[j,i,2]),gamma.los s[j]))

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#--- # Item-Loop, mixed gambles,gamble B

v.x.b[i,j] <- pow(prospects.b[j,i,1],alpha[j])

v.y.b[i,j] <- (-1 * lambda[j]) * pow((-1 * prospects.b[j,i,3]),alpha[j])

(delta.gain[j]*pow(prospects.b[j,i,2],gamma.gain[j])+pow((prospects.b[j,i,4]),gamma.gai n[j]))

w.y.b[i,j] <- delta.loss[j]*pow(prospects.b[j,i,4],gamma.loss[j])/

(delta.loss[j]*pow(prospects.b[j,i,4],gamma.loss[j])+pow((prospects.b[j,i,2]),gamma.los s[j]))

#---

# Item-Loop, mixed gambles,choice-rule

}

} }

Data Provided from Outside BUGS:

same as for the independent code

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Shrinkage for CPT Parameters

In resemblance to Figure 4, the plots display mean posterior estimates of the free parameters in CPT

separately for each individual at t1 and t2 (upper and lower row) and the hierarchically estimated parameters at t1 (middle row). For illustrative purposes, the points indicate a subset of 20 participants spaced out across the whole data range.

0.0 0.2 0.4 0.6 0.8 1.0

α

0.0 0.2 0.4 0.6 0.8 1.0

t2 individual t1 hierarchical t1 individual

0 1 2 3 4

δgain

0 1 2 3 4

0 1 2 3 4 5

δloss

0 1 2 3 4 5

0.0 0.2 0.4 0.6 0.8 1.0

γgain

0.0 0.2 0.4 0.6 0.8 1.0

γloss

0.0 0.2 0.4 0.6 0.8 1.0

0 1 2 3 4

λ

0 1 2 3 4

θ

0 1 2 3 4

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Shrinkage for TAX parameters

In resemblance to Figure 4, the plots display mean posterior estimates of the free parameters in TAX

separately for each individual at t1 and t2 (upper and lower row) and the hierarchically estimated parameters at t1 (middle row). For illustrative purposes, the points indicate a subset of 20 participants spaced out across the whole data range.

−2 −1 0 1 2

δ

−2 −1 0 1 2

t2 independent t1 hierarchical t1 independent

0 1 2 3 4 5

γ

0 1 2 3 4 5

0 1 2 3 4

θ

0 1 2 3 4